Rainbow Restrained Domination Numbers in Graphs

Abstract

A $2$-rainbow dominating function (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2\}$ such that for any vertex $v\in V(G)$ with $f(v)=\mathrm{\varnothing }$, the condition that there exists $u\in N(v)$ with $\bigcup _{u\in N(v)}f(u)=\{1,2\}$ is fulfilled, where $N(v)$ is the open neighborhood of $v$. A rainbow dominating function $f$ is said to be a rainbow restrained domination function if the induced subgraph of $G$ by the vertices with label $\mathrm{\varnothing }$ has no isolated vertex. The weight of a rainbow restrained dominating function is the value $w(f)=\sum _{u\in V(G)}|f(u)|$. The minimum weight of a rainbow restrained dominating function of $G$ is called the rainbow restrained domination number of $G$. In this paper, we initiate the study of the rainbow restrained domination number and we present some bounds for this parameter.